← Back to 2.1.2.3a1.3

ex.24.7.1.31_63_95.a

Base Field
\(F = \) 2.1.2.3a1.3 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 7\cdot 2 )\) View on LMFDB ↗
Description
exceptional, SL(2,3)
Construction
Extension to \(I_F\) of \( \tau = \operatorname{Ind}^{I_K}_{I_L} \chi \), with \(L, K\) and \(\chi\) as below
Semistability defect
\( e = 24\)
Conductor exponent
\( v(N) = 7\)
Character Order
4

Triply Field

The inertial type \(\tau\) becomes an induction (triply imprimitive) over: \( L = F(\mu_3, b) \), with \(\mu_3\) a root of \(x^2+x+1\) and \(b\) a root of \(x^{3} + a \)

Inducing Field

The inertial type \(\tau\) becomes reducible over \(K = L(c)\), \(c\) a root of \(x^{2} + 90546471444873528678335943241b^{3} x + (90546471444873528678335943241\mu_3 + 90546471444873528678335943241)b \)

Underlying Character

Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:

Order
4
Conductor exponent
11
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 11 })^\times/U_{\mathfrak{p}^{ 11 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} - 2a\cdot \mu_3)c + 4\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((2a\cdot \mu_3 + (2a + 2))b\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a + 4))b + a\cdot \mu_3 - 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b + 2)c + (3a\cdot \mu_3 + 3a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} - 2b + 4\mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 1 } \\ \chi^A\left((b + 2\mu_3)c + a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + (a + 2))b + ((-3a - 2)\mu_3 + 4))c + (3a + 3)b + (-3a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 2)\mu_3 + 4)b^{2} + (2a - 2)b - 3a\cdot \mu_3 - a)c + (-\mu_3 + (2a + 4))b^{2} + (-\mu_3 + (2a + 3))b + (3a - 3)\mu_3 - 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((2b^{2} - 2\mu_3b - 2a\cdot \mu_3 - 2a)\cdot c + 3b^{2} + (2a + 1)\mu_3b + (2a + 1)\mu_3 + 2a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 4)b^{2} + (2a + 4)b + 4a\cdot \mu_3)c + (2a + 2)\mu_3b^{2} + (\mu_3 + 4)b + (2a - 2)\mu_3 + 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (3a + 2))b^{2} + (-2\mu_3 + (3a + 2))b - 2\mu_3 + 4a)\cdot c + (2\mu_3 - 2)b^{2} + \mu_3b - 2\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((\mu_3b^{2} + (2\mu_3 + 2)b + 4)c + 2a\cdot \mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} + 2\mu_3)c + (4\mu_3 + 4)b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (484144114902291511791809789696a + 191615613701189410339093005440 )x^{47} + -602187528089655087873185232016a - 179129234276963203913858492624 x^{46} + (-393911808267358542931020755280a + 468173868907402891114887249568 )x^{45} + (335256493996062181962827572912a + 182440460880360006791421685152 )x^{44} + (256538990920345395342197476168a + 101156475274890974460123023176 )x^{43} + (610858259067452706881507611680a + 104908946903027723215123496692 )x^{42} + (-538684586000712822682933036264a + 202345010029789393824766691360 )x^{41} + 541529915616427994694834869564a - 543409694860391423816751353640 x^{40} + -326643034726057541718111418072a - 362417420136233080285809829088 x^{39} + (365734951293208613672588115360a + 37737339321036348182207314960 )x^{38} + (-549296162220500517024795627176a + 347356778847912448610709162832 )x^{37} + (-175889429835936284765625460136a + 460550192328625873586771415088 )x^{36} + 557317164062752833099494288688a - 212072539691623347153187890048 x^{35} + -300077620742433804272267236432a - 554791959513161370194587852256 x^{34} + 576106694165571489673409158512a - 169152339687358825995520958432 x^{33} + (-542952086824005225182146703204a + 523359523245486045525108991416 )x^{32} + 273534796315076433748699988704a - 485699924409677312032223073600 x^{31} + (-177950812762149027300357672112a + 84384933373626690767552069656 )x^{30} + (291213518601082744529749023144a + 489040702561488756662362642672 )x^{29} + (-235476515568888810561164594740a + 550014416547131546803284302940 )x^{28} + (447010663777629901956460271712a + 151021121877281580114080667848 )x^{27} + (-353221366629597332545362181016a + 48191163986068259152092569152 )x^{26} + (-192454005551327273129064795856a + 132778884220135136873586056672 )x^{25} + (629444272785046636483765320578a + 488399781827343120224505316096 )x^{24} + 313256009166556018986528091520a - 561351279873641379605217975776 x^{23} + -254277347655040164183244616056a - 449220752239672327256838984240 x^{22} + (-218613270607481085160249428272a + 239163176483337826797720933984 )x^{21} + (-223631705969497214712158007560a + 537993941984212471396755375824 )x^{20} + (-91075303320147400868697787288a + 79935768517326022752285692400 )x^{19} + 347592613080031627707683370888a - 442199068512400224829771832416 x^{18} + -97818507535948631614630354208a - 457273727814399248877800027872 x^{17} + (-404766827951025341025524595160a + 622619557514178607922551644632 )x^{16} + 302549489138897647628122024000a - 3830678180816631201567377456 x^{15} + -469005571407369207500276802856a - 267645031059684760662585833168 x^{14} + (136330613529835165989717287968a + 344279125820696904935764026944 )x^{13} + (244945795661485721935606936800a + 592454629546874993393734292652 )x^{12} + 133861330576336748584830192400a - 355419482788819304379065039392 x^{11} + (-549711745634470230239056301944a + 344204238511061054261249331672 )x^{10} + (466031225269415254070253311552a + 49515338118882697353234591472 )x^{9} + (-49559507946505555105908343576a + 112492774356498296847164823304 )x^{8} + (-276500548252847105454882843648a + 351693939997920116878368253024 )x^{7} + (269776602598105444037084621544a + 474208708087331681600304727536 )x^{6} + (418496709410009345795135077920a + 519859678918100341557522328256 )x^{5} + (224859534556900461738159432076a + 449090224121052332779794827048 )x^{4} + (-142604409785575245573799229760a + 100758344660212829701775364032 )x^{3} + -349118435834886830975754053744a - 99296791741636263405183982080 x^{2} + 78414564428263341908855304504a - 504668440066388893222526250160 x + 59883178486135861974544909952a + 414977993854584677284538644146 \)
← Back to 2.1.2.3a1.3 Summary